math.sqrt on complex, real part

Percentage Accurate: 41.4% → 82.9%

Time: 6.8s

Alternatives: 8

Speedup: 1.3×

Specification

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) + re)));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 41.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) + re)));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 82.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{+72}:\\ \;\;\;\;0.5 \cdot {\left({\left(e^{0.25}\right)}^{\left(\log \left(\frac{-1}{re}\right) + \log \left(im \cdot im\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(im, re\right) + re\right) \cdot 2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.1e+72)
   (* 0.5 (pow (pow (exp 0.25) (+ (log (/ -1.0 re)) (log (* im im)))) 2.0))
   (* (sqrt (* (+ (hypot im re) re) 2.0)) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.1e+72) {
		tmp = 0.5 * pow(pow(exp(0.25), (log((-1.0 / re)) + log((im * im)))), 2.0);
	} else {
		tmp = sqrt(((hypot(im, re) + re) * 2.0)) * 0.5;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.1e+72) {
		tmp = 0.5 * Math.pow(Math.pow(Math.exp(0.25), (Math.log((-1.0 / re)) + Math.log((im * im)))), 2.0);
	} else {
		tmp = Math.sqrt(((Math.hypot(im, re) + re) * 2.0)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.1e+72:
		tmp = 0.5 * math.pow(math.pow(math.exp(0.25), (math.log((-1.0 / re)) + math.log((im * im)))), 2.0)
	else:
		tmp = math.sqrt(((math.hypot(im, re) + re) * 2.0)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.1e+72)
		tmp = Float64(0.5 * ((exp(0.25) ^ Float64(log(Float64(-1.0 / re)) + log(Float64(im * im)))) ^ 2.0));
	else
		tmp = Float64(sqrt(Float64(Float64(hypot(im, re) + re) * 2.0)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.1e+72)
		tmp = 0.5 * ((exp(0.25) ^ (log((-1.0 / re)) + log((im * im)))) ^ 2.0);
	else
		tmp = sqrt(((hypot(im, re) + re) * 2.0)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.1e+72], N[(0.5 * N[Power[N[Power[N[Exp[0.25], $MachinePrecision], N[(N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision] + N[Log[N[(im * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision] + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.09999999999999988e72

   1. Initial program 6.6%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \cdot \frac{1}{2}} \]

      3. lower-*.f64 6.6

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \cdot 0.5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}} \cdot \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{re \cdot re + im \cdot im} + re\right) \cdot 2}} \cdot \frac{1}{2} \]

      6. lower-*.f64 6.6

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{re \cdot re + im \cdot im} + re\right) \cdot 2}} \cdot 0.5 \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\sqrt{re \cdot re + im \cdot im}} + re\right) \cdot 2} \cdot \frac{1}{2} \]

      8. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\sqrt{\color{blue}{re \cdot re + im \cdot im}} + re\right) \cdot 2} \cdot \frac{1}{2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(\sqrt{\color{blue}{im \cdot im + re \cdot re}} + re\right) \cdot 2} \cdot \frac{1}{2} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\sqrt{\color{blue}{im \cdot im} + re \cdot re} + re\right) \cdot 2} \cdot \frac{1}{2} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\sqrt{im \cdot im + \color{blue}{re \cdot re}} + re\right) \cdot 2} \cdot \frac{1}{2} \]

      12. lower-hypot.f64 37.2

          \[\leadsto \sqrt{\left(\color{blue}{\mathsf{hypot}\left(im, re\right)} + re\right) \cdot 2} \cdot 0.5 \]

   4. Applied rewrites 37.2%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{hypot}\left(im, re\right) + re\right) \cdot 2} \cdot 0.5} \]

   5. Taylor expanded in re around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot im}} \cdot \frac{1}{2} \]
   6. Step-by-step derivation

      1. lower-*.f64 6.9

         \[\leadsto \sqrt{\color{blue}{2 \cdot im}} \cdot 0.5 \]

   7. Applied rewrites 6.9%

      \[\leadsto \sqrt{\color{blue}{2 \cdot im}} \cdot 0.5 \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot im}} \cdot \frac{1}{2} \]

      2. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(2 \cdot im\right)}^{\frac{1}{2}}} \cdot \frac{1}{2} \]

      3. sqr-pow N/A

         \[\leadsto \color{blue}{\left({\left(2 \cdot im\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(2 \cdot im\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)} \cdot \frac{1}{2} \]

      4. pow2 N/A

         \[\leadsto \color{blue}{{\left({\left(2 \cdot im\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{2}} \cdot \frac{1}{2} \]

      5. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\left({\left(2 \cdot im\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{2}} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto {\color{blue}{\left({\left(2 \cdot im\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}}^{2} \cdot \frac{1}{2} \]

      7. metadata-eval N/A

         \[\leadsto {\left({\left(2 \cdot im\right)}^{\color{blue}{\frac{1}{4}}}\right)}^{2} \cdot \frac{1}{2} \]

   9. Applied rewrites 6.8%

      \[\leadsto \color{blue}{{\left({\left(2 \cdot im\right)}^{0.25}\right)}^{2}} \cdot 0.5 \]

   10. Taylor expanded in re around -inf

       \[\leadsto {\color{blue}{\left(e^{\frac{1}{4} \cdot \left(\log \left(\frac{-1}{re}\right) + \log \left({im}^{2}\right)\right)}\right)}}^{2} \cdot \frac{1}{2} \]
   11. Step-by-step derivation

       1. exp-prod N/A

          \[\leadsto {\color{blue}{\left({\left(e^{\frac{1}{4}}\right)}^{\left(\log \left(\frac{-1}{re}\right) + \log \left({im}^{2}\right)\right)}\right)}}^{2} \cdot \frac{1}{2} \]

       2. lower-pow.f64 N/A

          \[\leadsto {\color{blue}{\left({\left(e^{\frac{1}{4}}\right)}^{\left(\log \left(\frac{-1}{re}\right) + \log \left({im}^{2}\right)\right)}\right)}}^{2} \cdot \frac{1}{2} \]

       3. lower-exp.f64 N/A

          \[\leadsto {\left({\color{blue}{\left(e^{\frac{1}{4}}\right)}}^{\left(\log \left(\frac{-1}{re}\right) + \log \left({im}^{2}\right)\right)}\right)}^{2} \cdot \frac{1}{2} \]

       4. +-commutative N/A

          \[\leadsto {\left({\left(e^{\frac{1}{4}}\right)}^{\color{blue}{\left(\log \left({im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)}}\right)}^{2} \cdot \frac{1}{2} \]

       5. lower-+.f64 N/A

          \[\leadsto {\left({\left(e^{\frac{1}{4}}\right)}^{\color{blue}{\left(\log \left({im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)}}\right)}^{2} \cdot \frac{1}{2} \]

       6. lower-log.f64 N/A

          \[\leadsto {\left({\left(e^{\frac{1}{4}}\right)}^{\left(\color{blue}{\log \left({im}^{2}\right)} + \log \left(\frac{-1}{re}\right)\right)}\right)}^{2} \cdot \frac{1}{2} \]

       7. unpow2 N/A

          \[\leadsto {\left({\left(e^{\frac{1}{4}}\right)}^{\left(\log \color{blue}{\left(im \cdot im\right)} + \log \left(\frac{-1}{re}\right)\right)}\right)}^{2} \cdot \frac{1}{2} \]

       8. lower-*.f64 N/A

          \[\leadsto {\left({\left(e^{\frac{1}{4}}\right)}^{\left(\log \color{blue}{\left(im \cdot im\right)} + \log \left(\frac{-1}{re}\right)\right)}\right)}^{2} \cdot \frac{1}{2} \]

       9. lower-log.f64 N/A

          \[\leadsto {\left({\left(e^{\frac{1}{4}}\right)}^{\left(\log \left(im \cdot im\right) + \color{blue}{\log \left(\frac{-1}{re}\right)}\right)}\right)}^{2} \cdot \frac{1}{2} \]

       10. lower-/.f64 61.9

           \[\leadsto {\left({\left(e^{0.25}\right)}^{\left(\log \left(im \cdot im\right) + \log \color{blue}{\left(\frac{-1}{re}\right)}\right)}\right)}^{2} \cdot 0.5 \]

   12. Applied rewrites 61.9%

       \[\leadsto {\color{blue}{\left({\left(e^{0.25}\right)}^{\left(\log \left(im \cdot im\right) + \log \left(\frac{-1}{re}\right)\right)}\right)}}^{2} \cdot 0.5 \]

   if -3.09999999999999988e72 < re

   1. Initial program 47.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \cdot \frac{1}{2}} \]

      3. lower-*.f64 47.9

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \cdot 0.5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}} \cdot \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{re \cdot re + im \cdot im} + re\right) \cdot 2}} \cdot \frac{1}{2} \]

      6. lower-*.f64 47.9

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{re \cdot re + im \cdot im} + re\right) \cdot 2}} \cdot 0.5 \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\sqrt{re \cdot re + im \cdot im}} + re\right) \cdot 2} \cdot \frac{1}{2} \]

      8. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\sqrt{\color{blue}{re \cdot re + im \cdot im}} + re\right) \cdot 2} \cdot \frac{1}{2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(\sqrt{\color{blue}{im \cdot im + re \cdot re}} + re\right) \cdot 2} \cdot \frac{1}{2} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\sqrt{\color{blue}{im \cdot im} + re \cdot re} + re\right) \cdot 2} \cdot \frac{1}{2} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\sqrt{im \cdot im + \color{blue}{re \cdot re}} + re\right) \cdot 2} \cdot \frac{1}{2} \]

      12. lower-hypot.f64 91.5

          \[\leadsto \sqrt{\left(\color{blue}{\mathsf{hypot}\left(im, re\right)} + re\right) \cdot 2} \cdot 0.5 \]

   4. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{hypot}\left(im, re\right) + re\right) \cdot 2} \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{+72}:\\ \;\;\;\;0.5 \cdot {\left({\left(e^{0.25}\right)}^{\left(\log \left(\frac{-1}{re}\right) + \log \left(im \cdot im\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(im, re\right) + re\right) \cdot 2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.2 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{\frac{\frac{im}{re}}{\frac{-1}{im}}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(im, re\right) + re\right) \cdot 2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.2e+76)
   (* (sqrt (/ (/ im re) (/ -1.0 im))) 0.5)
   (* (sqrt (* (+ (hypot im re) re) 2.0)) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.2e+76) {
		tmp = sqrt(((im / re) / (-1.0 / im))) * 0.5;
	} else {
		tmp = sqrt(((hypot(im, re) + re) * 2.0)) * 0.5;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.2e+76) {
		tmp = Math.sqrt(((im / re) / (-1.0 / im))) * 0.5;
	} else {
		tmp = Math.sqrt(((Math.hypot(im, re) + re) * 2.0)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.2e+76:
		tmp = math.sqrt(((im / re) / (-1.0 / im))) * 0.5
	else:
		tmp = math.sqrt(((math.hypot(im, re) + re) * 2.0)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.2e+76)
		tmp = Float64(sqrt(Float64(Float64(im / re) / Float64(-1.0 / im))) * 0.5);
	else
		tmp = Float64(sqrt(Float64(Float64(hypot(im, re) + re) * 2.0)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.2e+76)
		tmp = sqrt(((im / re) / (-1.0 / im))) * 0.5;
	else
		tmp = sqrt(((hypot(im, re) + re) * 2.0)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.2e+76], N[(N[Sqrt[N[(N[(im / re), $MachinePrecision] / N[(-1.0 / im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Sqrt[N[(N[(N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision] + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.20000000000000013e76

   1. Initial program 6.6%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\mathsf{neg}\left(\frac{{im}^{2}}{re}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{neg}\left(\frac{\color{blue}{im \cdot im}}{re}\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{neg}\left(\color{blue}{im \cdot \frac{im}{re}}\right)} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(im\right)\right) \cdot \frac{im}{re}}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(-1 \cdot im\right)} \cdot \frac{im}{re}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(-1 \cdot im\right) \cdot \frac{im}{re}}} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(im\right)\right)} \cdot \frac{im}{re}} \]

      8. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(-im\right)} \cdot \frac{im}{re}} \]

      9. lower-/.f64 57.4

         \[\leadsto 0.5 \cdot \sqrt{\left(-im\right) \cdot \color{blue}{\frac{im}{re}}} \]

   5. Applied rewrites 57.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(-im\right) \cdot \frac{im}{re}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 53.9%

      \[\leadsto 0.5 \cdot \sqrt{\frac{\left(-im\right) \cdot im}{\color{blue}{re}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 57.4%

      \[\leadsto 0.5 \cdot \sqrt{\frac{\frac{im}{re}}{\color{blue}{\frac{-1}{im}}}} \]

   if -4.20000000000000013e76 < re

   1. Initial program 47.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \cdot \frac{1}{2}} \]

      3. lower-*.f64 47.9

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \cdot 0.5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}} \cdot \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{re \cdot re + im \cdot im} + re\right) \cdot 2}} \cdot \frac{1}{2} \]

      6. lower-*.f64 47.9

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{re \cdot re + im \cdot im} + re\right) \cdot 2}} \cdot 0.5 \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\sqrt{re \cdot re + im \cdot im}} + re\right) \cdot 2} \cdot \frac{1}{2} \]

      8. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\sqrt{\color{blue}{re \cdot re + im \cdot im}} + re\right) \cdot 2} \cdot \frac{1}{2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(\sqrt{\color{blue}{im \cdot im + re \cdot re}} + re\right) \cdot 2} \cdot \frac{1}{2} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\sqrt{\color{blue}{im \cdot im} + re \cdot re} + re\right) \cdot 2} \cdot \frac{1}{2} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\sqrt{im \cdot im + \color{blue}{re \cdot re}} + re\right) \cdot 2} \cdot \frac{1}{2} \]

      12. lower-hypot.f64 91.5

          \[\leadsto \sqrt{\left(\color{blue}{\mathsf{hypot}\left(im, re\right)} + re\right) \cdot 2} \cdot 0.5 \]

   4. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{hypot}\left(im, re\right) + re\right) \cdot 2} \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.2 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{\frac{\frac{im}{re}}{\frac{-1}{im}}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(im, re\right) + re\right) \cdot 2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{\frac{\frac{im}{re}}{\frac{-1}{im}}} \cdot 0.5\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(im + re, 2, \frac{re}{im} \cdot re\right)} \cdot 0.5\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.1e+72)
   (* (sqrt (/ (/ im re) (/ -1.0 im))) 0.5)
   (if (<= re 2.5e-117)
     (* (sqrt (fma (+ im re) 2.0 (* (/ re im) re))) 0.5)
     (if (<= re 7e+118)
       (* (sqrt (* (+ (sqrt (fma re re (* im im))) re) 2.0)) 0.5)
       (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.1e+72) {
		tmp = sqrt(((im / re) / (-1.0 / im))) * 0.5;
	} else if (re <= 2.5e-117) {
		tmp = sqrt(fma((im + re), 2.0, ((re / im) * re))) * 0.5;
	} else if (re <= 7e+118) {
		tmp = sqrt(((sqrt(fma(re, re, (im * im))) + re) * 2.0)) * 0.5;
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.1e+72)
		tmp = Float64(sqrt(Float64(Float64(im / re) / Float64(-1.0 / im))) * 0.5);
	elseif (re <= 2.5e-117)
		tmp = Float64(sqrt(fma(Float64(im + re), 2.0, Float64(Float64(re / im) * re))) * 0.5);
	elseif (re <= 7e+118)
		tmp = Float64(sqrt(Float64(Float64(sqrt(fma(re, re, Float64(im * im))) + re) * 2.0)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.1e+72], N[(N[Sqrt[N[(N[(im / re), $MachinePrecision] / N[(-1.0 / im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 2.5e-117], N[(N[Sqrt[N[(N[(im + re), $MachinePrecision] * 2.0 + N[(N[(re / im), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 7e+118], N[(N[Sqrt[N[(N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3.09999999999999988e72

   1. Initial program 6.6%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\mathsf{neg}\left(\frac{{im}^{2}}{re}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{neg}\left(\frac{\color{blue}{im \cdot im}}{re}\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{neg}\left(\color{blue}{im \cdot \frac{im}{re}}\right)} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(im\right)\right) \cdot \frac{im}{re}}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(-1 \cdot im\right)} \cdot \frac{im}{re}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(-1 \cdot im\right) \cdot \frac{im}{re}}} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(im\right)\right)} \cdot \frac{im}{re}} \]

      8. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(-im\right)} \cdot \frac{im}{re}} \]

      9. lower-/.f64 57.4

         \[\leadsto 0.5 \cdot \sqrt{\left(-im\right) \cdot \color{blue}{\frac{im}{re}}} \]

   5. Applied rewrites 57.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(-im\right) \cdot \frac{im}{re}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 53.9%

      \[\leadsto 0.5 \cdot \sqrt{\frac{\left(-im\right) \cdot im}{\color{blue}{re}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 57.4%

      \[\leadsto 0.5 \cdot \sqrt{\frac{\frac{im}{re}}{\color{blue}{\frac{-1}{im}}}} \]

   if -3.09999999999999988e72 < re < 2.5e-117

   1. Initial program 45.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{2 \cdot im + re \cdot \left(2 + \frac{re}{im}\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot \left(2 + \frac{re}{im}\right) + 2 \cdot im}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(2 + \frac{re}{im}\right) \cdot re} + 2 \cdot im} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(2 + \frac{re}{im}, re, 2 \cdot im\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\frac{re}{im} + 2}, re, 2 \cdot im\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\frac{re}{im} + 2}, re, 2 \cdot im\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\frac{re}{im}} + 2, re, 2 \cdot im\right)} \]

      7. lower-*.f64 41.2

         \[\leadsto 0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im} + 2, re, \color{blue}{2 \cdot im}\right)} \]

   5. Applied rewrites 41.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{re}{im} + 2, re, 2 \cdot im\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 41.2%

      \[\leadsto 0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im}, \color{blue}{re}, \mathsf{fma}\left(2, re, 2 \cdot im\right)\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im}, re, \mathsf{fma}\left(2, re, 2 \cdot im\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\frac{re}{im}, re, \mathsf{fma}\left(2, re, 2 \cdot im\right)\right)} \cdot \frac{1}{2}} \]

      3. lower-*.f64 41.2

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\frac{re}{im}, re, \mathsf{fma}\left(2, re, 2 \cdot im\right)\right)} \cdot 0.5} \]

   9. Applied rewrites 41.2%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(re + im, 2, \frac{re}{im} \cdot re\right)} \cdot 0.5} \]

   if 2.5e-117 < re < 7.00000000000000033e118

   1. Initial program 80.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{re \cdot re + im \cdot im}} + re\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{re \cdot re} + im \cdot im} + re\right)} \]

      3. lower-fma.f64 80.8

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(re, re, im \cdot im\right)}} + re\right)} \]

   4. Applied rewrites 80.8%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(re, re, im \cdot im\right)}} + re\right)} \]

   if 7.00000000000000033e118 < re

   1. Initial program 11.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({\left(\sqrt{2}\right)}^{2} \cdot \sqrt{re}\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \sqrt{re}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{2} \cdot \sqrt{re}\right) \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot 2\right) \cdot \sqrt{re}} \]

      5. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot \sqrt{re} \]

      6. *-lft-identity N/A

         \[\leadsto \color{blue}{\sqrt{re}} \]

      7. lower-sqrt.f64 90.2

         \[\leadsto \color{blue}{\sqrt{re}} \]

   5. Applied rewrites 90.2%

      \[\leadsto \color{blue}{\sqrt{re}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{\frac{\frac{im}{re}}{\frac{-1}{im}}} \cdot 0.5\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(im + re, 2, \frac{re}{im} \cdot re\right)} \cdot 0.5\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(im + re, 2, \frac{re}{im} \cdot re\right)} \cdot 0.5\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.1e+72)
   (* (sqrt (* (/ (- im) re) im)) 0.5)
   (if (<= re 2.5e-117)
     (* (sqrt (fma (+ im re) 2.0 (* (/ re im) re))) 0.5)
     (if (<= re 7e+118)
       (* (sqrt (* (+ (sqrt (fma re re (* im im))) re) 2.0)) 0.5)
       (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.1e+72) {
		tmp = sqrt(((-im / re) * im)) * 0.5;
	} else if (re <= 2.5e-117) {
		tmp = sqrt(fma((im + re), 2.0, ((re / im) * re))) * 0.5;
	} else if (re <= 7e+118) {
		tmp = sqrt(((sqrt(fma(re, re, (im * im))) + re) * 2.0)) * 0.5;
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.1e+72)
		tmp = Float64(sqrt(Float64(Float64(Float64(-im) / re) * im)) * 0.5);
	elseif (re <= 2.5e-117)
		tmp = Float64(sqrt(fma(Float64(im + re), 2.0, Float64(Float64(re / im) * re))) * 0.5);
	elseif (re <= 7e+118)
		tmp = Float64(sqrt(Float64(Float64(sqrt(fma(re, re, Float64(im * im))) + re) * 2.0)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.1e+72], N[(N[Sqrt[N[(N[((-im) / re), $MachinePrecision] * im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 2.5e-117], N[(N[Sqrt[N[(N[(im + re), $MachinePrecision] * 2.0 + N[(N[(re / im), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 7e+118], N[(N[Sqrt[N[(N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3.09999999999999988e72

   1. Initial program 6.6%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\mathsf{neg}\left(\frac{{im}^{2}}{re}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{neg}\left(\frac{\color{blue}{im \cdot im}}{re}\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{neg}\left(\color{blue}{im \cdot \frac{im}{re}}\right)} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(im\right)\right) \cdot \frac{im}{re}}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(-1 \cdot im\right)} \cdot \frac{im}{re}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(-1 \cdot im\right) \cdot \frac{im}{re}}} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(im\right)\right)} \cdot \frac{im}{re}} \]

      8. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(-im\right)} \cdot \frac{im}{re}} \]

      9. lower-/.f64 57.4

         \[\leadsto 0.5 \cdot \sqrt{\left(-im\right) \cdot \color{blue}{\frac{im}{re}}} \]

   5. Applied rewrites 57.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(-im\right) \cdot \frac{im}{re}}} \]

   if -3.09999999999999988e72 < re < 2.5e-117

   1. Initial program 45.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{2 \cdot im + re \cdot \left(2 + \frac{re}{im}\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot \left(2 + \frac{re}{im}\right) + 2 \cdot im}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(2 + \frac{re}{im}\right) \cdot re} + 2 \cdot im} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(2 + \frac{re}{im}, re, 2 \cdot im\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\frac{re}{im} + 2}, re, 2 \cdot im\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\frac{re}{im} + 2}, re, 2 \cdot im\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\frac{re}{im}} + 2, re, 2 \cdot im\right)} \]

      7. lower-*.f64 41.2

         \[\leadsto 0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im} + 2, re, \color{blue}{2 \cdot im}\right)} \]

   5. Applied rewrites 41.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{re}{im} + 2, re, 2 \cdot im\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 41.2%

      \[\leadsto 0.5 \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im}, \color{blue}{re}, \mathsf{fma}\left(2, re, 2 \cdot im\right)\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\mathsf{fma}\left(\frac{re}{im}, re, \mathsf{fma}\left(2, re, 2 \cdot im\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\frac{re}{im}, re, \mathsf{fma}\left(2, re, 2 \cdot im\right)\right)} \cdot \frac{1}{2}} \]

      3. lower-*.f64 41.2

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\frac{re}{im}, re, \mathsf{fma}\left(2, re, 2 \cdot im\right)\right)} \cdot 0.5} \]

   9. Applied rewrites 41.2%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(re + im, 2, \frac{re}{im} \cdot re\right)} \cdot 0.5} \]

   if 2.5e-117 < re < 7.00000000000000033e118

   1. Initial program 80.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{re \cdot re + im \cdot im}} + re\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{re \cdot re} + im \cdot im} + re\right)} \]

      3. lower-fma.f64 80.8

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(re, re, im \cdot im\right)}} + re\right)} \]

   4. Applied rewrites 80.8%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(re, re, im \cdot im\right)}} + re\right)} \]

   if 7.00000000000000033e118 < re

   1. Initial program 11.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({\left(\sqrt{2}\right)}^{2} \cdot \sqrt{re}\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \sqrt{re}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{2} \cdot \sqrt{re}\right) \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot 2\right) \cdot \sqrt{re}} \]

      5. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot \sqrt{re} \]

      6. *-lft-identity N/A

         \[\leadsto \color{blue}{\sqrt{re}} \]

      7. lower-sqrt.f64 90.2

         \[\leadsto \color{blue}{\sqrt{re}} \]

   5. Applied rewrites 90.2%

      \[\leadsto \color{blue}{\sqrt{re}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(im + re, 2, \frac{re}{im} \cdot re\right)} \cdot 0.5\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{elif}\;re \leq 8.8 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{\left(im + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.1e+72)
   (* (sqrt (* (/ (- im) re) im)) 0.5)
   (if (<= re 8.8e+17) (* (sqrt (* (+ im re) 2.0)) 0.5) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.1e+72) {
		tmp = sqrt(((-im / re) * im)) * 0.5;
	} else if (re <= 8.8e+17) {
		tmp = sqrt(((im + re) * 2.0)) * 0.5;
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.1d+72)) then
        tmp = sqrt(((-im / re) * im)) * 0.5d0
    else if (re <= 8.8d+17) then
        tmp = sqrt(((im + re) * 2.0d0)) * 0.5d0
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.1e+72) {
		tmp = Math.sqrt(((-im / re) * im)) * 0.5;
	} else if (re <= 8.8e+17) {
		tmp = Math.sqrt(((im + re) * 2.0)) * 0.5;
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.1e+72:
		tmp = math.sqrt(((-im / re) * im)) * 0.5
	elif re <= 8.8e+17:
		tmp = math.sqrt(((im + re) * 2.0)) * 0.5
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.1e+72)
		tmp = Float64(sqrt(Float64(Float64(Float64(-im) / re) * im)) * 0.5);
	elseif (re <= 8.8e+17)
		tmp = Float64(sqrt(Float64(Float64(im + re) * 2.0)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.1e+72)
		tmp = sqrt(((-im / re) * im)) * 0.5;
	elseif (re <= 8.8e+17)
		tmp = sqrt(((im + re) * 2.0)) * 0.5;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.1e+72], N[(N[Sqrt[N[(N[((-im) / re), $MachinePrecision] * im), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 8.8e+17], N[(N[Sqrt[N[(N[(im + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.09999999999999988e72

   1. Initial program 6.6%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\mathsf{neg}\left(\frac{{im}^{2}}{re}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{neg}\left(\frac{\color{blue}{im \cdot im}}{re}\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\mathsf{neg}\left(\color{blue}{im \cdot \frac{im}{re}}\right)} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(im\right)\right) \cdot \frac{im}{re}}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(-1 \cdot im\right)} \cdot \frac{im}{re}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(-1 \cdot im\right) \cdot \frac{im}{re}}} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(im\right)\right)} \cdot \frac{im}{re}} \]

      8. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(-im\right)} \cdot \frac{im}{re}} \]

      9. lower-/.f64 57.4

         \[\leadsto 0.5 \cdot \sqrt{\left(-im\right) \cdot \color{blue}{\frac{im}{re}}} \]

   5. Applied rewrites 57.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(-im\right) \cdot \frac{im}{re}}} \]

   if -3.09999999999999988e72 < re < 8.8e17

   1. Initial program 50.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{\left(im + re\right)}} \]
   4. Step-by-step derivation

      1. lower-+.f64 39.6

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(im + re\right)}} \]

   5. Applied rewrites 39.6%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(im + re\right)}} \]

   if 8.8e17 < re

   1. Initial program 42.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({\left(\sqrt{2}\right)}^{2} \cdot \sqrt{re}\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \sqrt{re}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{2} \cdot \sqrt{re}\right) \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot 2\right) \cdot \sqrt{re}} \]

      5. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot \sqrt{re} \]

      6. *-lft-identity N/A

         \[\leadsto \color{blue}{\sqrt{re}} \]

      7. lower-sqrt.f64 83.8

         \[\leadsto \color{blue}{\sqrt{re}} \]

   5. Applied rewrites 83.8%

      \[\leadsto \color{blue}{\sqrt{re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{\frac{-im}{re} \cdot im} \cdot 0.5\\ \mathbf{elif}\;re \leq 8.8 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{\left(im + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.2 \cdot 10^{+219}:\\ \;\;\;\;\sqrt{\left(\left(-re\right) + re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \leq 8.8 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{\left(im + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.2e+219)
   (* (sqrt (* (+ (- re) re) 2.0)) 0.5)
   (if (<= re 8.8e+17) (* (sqrt (* (+ im re) 2.0)) 0.5) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.2e+219) {
		tmp = sqrt(((-re + re) * 2.0)) * 0.5;
	} else if (re <= 8.8e+17) {
		tmp = sqrt(((im + re) * 2.0)) * 0.5;
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-6.2d+219)) then
        tmp = sqrt(((-re + re) * 2.0d0)) * 0.5d0
    else if (re <= 8.8d+17) then
        tmp = sqrt(((im + re) * 2.0d0)) * 0.5d0
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -6.2e+219) {
		tmp = Math.sqrt(((-re + re) * 2.0)) * 0.5;
	} else if (re <= 8.8e+17) {
		tmp = Math.sqrt(((im + re) * 2.0)) * 0.5;
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6.2e+219:
		tmp = math.sqrt(((-re + re) * 2.0)) * 0.5
	elif re <= 8.8e+17:
		tmp = math.sqrt(((im + re) * 2.0)) * 0.5
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.2e+219)
		tmp = Float64(sqrt(Float64(Float64(Float64(-re) + re) * 2.0)) * 0.5);
	elseif (re <= 8.8e+17)
		tmp = Float64(sqrt(Float64(Float64(im + re) * 2.0)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -6.2e+219)
		tmp = sqrt(((-re + re) * 2.0)) * 0.5;
	elseif (re <= 8.8e+17)
		tmp = sqrt(((im + re) * 2.0)) * 0.5;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6.2e+219], N[(N[Sqrt[N[(N[((-re) + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[re, 8.8e+17], N[(N[Sqrt[N[(N[(im + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -6.19999999999999938e219

   1. Initial program 2.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\color{blue}{-1 \cdot re} + re\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\mathsf{neg}\left(re\right)\right)} + re\right)} \]

      2. lower-neg.f64 34.6

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(-re\right)} + re\right)} \]

   5. Applied rewrites 34.6%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(-re\right)} + re\right)} \]

   if -6.19999999999999938e219 < re < 8.8e17

   1. Initial program 42.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{\left(im + re\right)}} \]
   4. Step-by-step derivation

      1. lower-+.f64 34.1

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(im + re\right)}} \]

   5. Applied rewrites 34.1%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(im + re\right)}} \]

   if 8.8e17 < re

   1. Initial program 42.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({\left(\sqrt{2}\right)}^{2} \cdot \sqrt{re}\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \sqrt{re}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{2} \cdot \sqrt{re}\right) \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot 2\right) \cdot \sqrt{re}} \]

      5. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot \sqrt{re} \]

      6. *-lft-identity N/A

         \[\leadsto \color{blue}{\sqrt{re}} \]

      7. lower-sqrt.f64 83.8

         \[\leadsto \color{blue}{\sqrt{re}} \]

   5. Applied rewrites 83.8%

      \[\leadsto \color{blue}{\sqrt{re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.2 \cdot 10^{+219}:\\ \;\;\;\;\sqrt{\left(\left(-re\right) + re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \leq 8.8 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{\left(im + re\right) \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 40.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 43000000000000:\\ \;\;\;\;\sqrt{im \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 43000000000000.0) (* (sqrt (* im 2.0)) 0.5) (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 43000000000000.0) {
		tmp = sqrt((im * 2.0)) * 0.5;
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 43000000000000.0d0) then
        tmp = sqrt((im * 2.0d0)) * 0.5d0
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 43000000000000.0) {
		tmp = Math.sqrt((im * 2.0)) * 0.5;
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 43000000000000.0:
		tmp = math.sqrt((im * 2.0)) * 0.5
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 43000000000000.0)
		tmp = Float64(sqrt(Float64(im * 2.0)) * 0.5);
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 43000000000000.0)
		tmp = sqrt((im * 2.0)) * 0.5;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 43000000000000.0], N[(N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.3e13

   1. Initial program 37.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{2 \cdot im}} \]
   4. Step-by-step derivation

      1. lower-*.f64 29.1

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]

   5. Applied rewrites 29.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{2 \cdot im}} \]

   if 4.3e13 < re

   1. Initial program 42.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({\left(\sqrt{2}\right)}^{2} \cdot \sqrt{re}\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \sqrt{re}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{2} \cdot \sqrt{re}\right) \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot 2\right) \cdot \sqrt{re}} \]

      5. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot \sqrt{re} \]

      6. *-lft-identity N/A

         \[\leadsto \color{blue}{\sqrt{re}} \]

      7. lower-sqrt.f64 83.8

         \[\leadsto \color{blue}{\sqrt{re}} \]

   5. Applied rewrites 83.8%

      \[\leadsto \color{blue}{\sqrt{re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 43000000000000:\\ \;\;\;\;\sqrt{im \cdot 2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 26.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{re} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (sqrt re))

C

double code(double re, double im) {
	return sqrt(re);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sqrt(re)
end function

Java

public static double code(double re, double im) {
	return Math.sqrt(re);
}

Python

def code(re, im):
	return math.sqrt(re)

Julia

function code(re, im)
	return sqrt(re)
end

MATLAB

function tmp = code(re, im)
	tmp = sqrt(re);
end

Wolfram

code[re_, im_] := N[Sqrt[re], $MachinePrecision]

Derivation

1. Initial program 39.0%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
2. Add Preprocessing

3. Taylor expanded in re around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({\left(\sqrt{2}\right)}^{2} \cdot \sqrt{re}\right)} \]

   2. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \sqrt{re}\right) \]

   3. rem-square-sqrt N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{2} \cdot \sqrt{re}\right) \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot 2\right) \cdot \sqrt{re}} \]

   5. metadata-eval N/A

      \[\leadsto \color{blue}{1} \cdot \sqrt{re} \]

   6. *-lft-identity N/A

      \[\leadsto \color{blue}{\sqrt{re}} \]

   7. lower-sqrt.f64 27.1

      \[\leadsto \color{blue}{\sqrt{re}} \]

5. Applied rewrites 27.1%

   \[\leadsto \color{blue}{\sqrt{re}} \]
6. Add Preprocessing

Developer Target 1: 49.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{t\_0 - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(t\_0 + re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* re re) (* im im)))))
   (if (< re 0.0)
     (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- t_0 re)))))
     (* 0.5 (sqrt (* 2.0 (+ t_0 re)))))))

C

double code(double re, double im) {
	double t_0 = sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((re * re) + (im * im)))
    if (re < 0.0d0) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(((im * im) / (t_0 - re))))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (t_0 + re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sqrt(((re * re) + (im * im)))
	tmp = 0
	if re < 0.0:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt(((im * im) / (t_0 - re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (t_0 + re)))
	return tmp

Julia

function code(re, im)
	t_0 = sqrt(Float64(Float64(re * re) + Float64(im * im)))
	tmp = 0.0
	if (re < 0.0)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(Float64(Float64(im * im) / Float64(t_0 - re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(t_0 + re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt(((re * re) + (im * im)));
	tmp = 0.0;
	if (re < 0.0)
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	else
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[re, 0.0], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(im * im), $MachinePrecision] / N[(t$95$0 - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(t$95$0 + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024325 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< re 0) (* 1/2 (* (sqrt 2) (sqrt (/ (* im im) (- (modulus re im) re))))) (* 1/2 (sqrt (* 2 (+ (modulus re im) re))))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))